1
One (1) is a positive integer one more than 0 and one less than 2. Properties One is the only positive integer that is neither a prime number nor a composite number. Any number multiplied by itself is the other number, that is, 1*n = n. Any number to the power of 1 is itself, or in other words, n^1 = n. And one to any power is 1, so 1^n = 1. One is the first in many sequences. It is the first square number (next is 4), cube number (next is 8), Fibonacci number (also the second one, next is 2), triangular number (next is 3), Lucas number (next is 3), tetrahedral number (next is 4), factorial number (next is 2), square pyramidal number (next is 5), pentatope number (next is 5), lucky number (next is 3), highly composite number (next is 2), deficient number, fourth power (next is 16), and many, many more. Many of these properties are trivial. It is also the only number that can be divided into any integer, as x/1 = x. Logarithms on base 1 are undefined, as 1^x is always 1. The 1st root of a number is always just the original number. Properties 1 is the first natural number. 1 is , and it's the only natural number that is neither prime nor composite. 1 is a , , , etc. A , cubic number, etc. 1 is the multiplicative identity, meaning that \(a = a \times 1\) for all \(a\). In fact, \(a \underbrace{\uparrow\uparrow\ldots\uparrow\uparrow}_n 1 = a\) (arrow notation) for all \(n,a \geq 0\), so 1 is a sort of identity for all the hyper operators beyond addition. Furthermore, for all \(n,a > 0\), \(a \underbrace{\uparrow\uparrow\ldots\uparrow\uparrow}_n 0 = 1\). 1 appears frequently as a "default" argument in googological notations, such as BEAF, chained arrow notation, and hyper-E notation. By definition all natural numbers are just strings of 1's added together. For example, 1,000,000 is a string of 1 million 1's added together. In googology Sbiis Saibian argued that all numbers larger than 1 should be called "large numbers," because the reciprocals of large numbers are small. The large numbers and small numbers are "mirrored" about 1, so it makes sense to say that 1 is the threshold of largeness. A number like 1 + 1/googol could be called a "very small large number." The smallest large number, obviously, also cannot exist. Wiki user Donald Knuth calls this number miad, and it's equal to using the -yllion system. 1 was also the first number by Adam Elga in the Big Number Duel. 1 can be named garone, fzone, fugaone, megafugaone, oneplaton, and onesuplaton with the gar-, fz-, fuga-, megafuga-, -platon, and -suplaton prefixes respectively. Googological functions returning 1 *Rado's Sigma Function: \(\Sigma(1)=1\) *Maximum shifts function: \(S(1)=1\) *Xi function: \(\Xi(1)=1\) *Goodstein function: \(G(1)=1\) *Weak Goodstein function: \(g(1)=1\) *Kirby-Paris hydra: \(\text{Hydra}(1)=1\) *Buchholz hydra: \(\text{BH}(2)=1\) *TREE function: \(\text{TREE}(1)=1\) *Weak tree function: \(\text{tree}(0)=1\) *Fusible numbers: \(m_1(0) = 1\) *Exploding Tree function: \(E(1)=1\) *Latin square: \(L(1)=1\) *Gijswijt's sequence: \(c(1)=1\) As a cash denomination Some currencies, such as the and the , have banknotes with this number in the denomination. Some currencies, such as the and the , have coins with this number in the denomination. Sources